People measure project success on a number of criteria, and some of them are totally unrelated to each other. There are both subjective and objective criteria, and the people who assess the criteria will vary in authority and power from project to project. Many inter-organisational projects have their success factors written down and are checked against specific, measurable criteria prior to payments or approvals. Is this enough to make your project a success?
As a project manager you simply have to address any written and specific success criteria. But you also need to look beyond the written or spoken requirements, which are often the minimum success criteria and look to the client's experience on the project versus their expectations, needs and wants. There are often hidden or unspoken agendas that need to be satisfied. You need to speak with not just the sponsor, but all the key stakeholders, influencers. and any proxies for those stakeholders and understand what experiences they are looking for from your project. Some will come from the product or service that is being produced and others will come from the experience of participating in the project.
A typical example of the way people's requirements come in layers is the way problems are handled. Some sponsors want to hear about any significant problem as soon as it occurs and others just want to hear from you when the project is completed with a turnkey solution. Some are looking to impress their peers with a big budget production, while others want a cheap and quick solution to their business problem.
Another example, and one of the most common problems in large corporate project environments, is the tension between development cost and time versus the IT department;'s desire to keep their infrastructure secure and manageable. If you don't address this, often unspoken, requirement early on you and your stakeholders will no doubt be wondering what hit them when the IT time and costs estimates come in.
You need to communicate with your stakeholders and know what experiences they are looking for and manage to both sets of requirements. To do this you'll need to know both who your stakeholders are and what their agendas are.
At this point I'll refer to the Tom Peters Brand You concept and to Noriaki Kano's ideas about understanding requirements for the "how to" bit; you need to use your interpersonal skills in getting people to trust you enough to tell you what their real hopes and fears are, and discuss what you can do for them. When discussing their requirements people often focus on the functional must haves (in the parlance of marketers - the steak) - but you need to step beyond those and find out what will really get people excited from your project (the sizzle.)
And then re-inforce Brand You by doing what it takes to deliver on your promises.